Pulse compression during second-harmonic generation in engineered aperiodic quasi-phase-matching gratings

We theoretically propose a new procedure of designing quasi-phase-matching (QPM) gratings for compressing optical ultra-short pulses during second-harmonic generation. The grating consists of blocks of crystal with same block length and the direction of spontaneous polarization of each block is determined by optimal algorithm, by which the sign of the nonlinear coefficient of each block is optimized to make the phase response of the grating same for different wavelength at second harmonic waves, so that the generated second harmonic pulse from the end of the crystal will be compressed.during nonlinear optical frequency conversion process. ©2005 Optical Society of America OCIS codes: (320.5520) Pulse compression; (140.7090) Ultrafast lasers; (190.4360) Nonlinear optics devices; (190.2620) Frequency conversion; References and links 1. J.A. Giordmaine, M.A. Duguay,and J.W. Hansen, “Compression of optical pulses (Mode locked HeNe laser generated light pulse compression in time without energy loss, using method similar to chirp radar method),” IEEE J. Quantum Electron. QE-4 252 (1968). 2. D.Strickland and G.. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219 (1985). 3. M.A. Arbore, O. Marco, and M.M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett. 22, 865 (1997) 4. M.A. Arbore, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341 (1997) 5. G.. Imeshev, “Engineerable femtosecond pulse shaping by second-harmonic generation with Fourier synthetic quasi-phase-matching gratings,” Opt. Lett. 23, 864 (1998) 6. L. Gallmann, G.. Steinmeyer, and U. Keller, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings,” Opt. Lett. 26, 614 (2001) 7. G..I meshev, M.A. Arbore,*and M.M. Fejer, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B 17, 304 (2000) 8. G.. Imeshev, M.A. Arbore,*and M.M. Fejer, “Pulse shaping and compression by second-harmonic generation with quasi-phase-matching gratings in the presence of arbitrary dispersion,” J. Opt. Soc. Am. B 17, 1420 (2000) 9. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. 22, 1553 (1997)


Introduction
Optical pulse compression, 1 applied in many ultrafast laser systems, has become increasingly important since the development of chirped pulse amplification.conversion, especially second-harmonic generation, has been used for pulse compression, [3][4][5][6] extending the wavelength range that is accessible to ultrafast laser systems.It was shown that a chirped-grating quasi-phase-matched (QPM) second-harmonic (SH) generator can provide significant effective dispersion at the SH wavelength relative to the fundamental wavelength, which eliminating(s) the need for dispersive delay lines in some ultrafast laser system 3-6 .These monolithic devices are far more compact than diffraction-grating-based or prism-based dispersive delay lines and offer high power-handling capability.Pulse compression with longitudinally nonuniform QPM gratings, can be most intuitively understood in the time domain.For chirped fundamental (input) pulses, frequency components correspond to temporal slices.These temporal slices are frequency doubled at positions in the material where the grating period quasi-phase matches the interaction.By choosing the location of each spatial frequency component of the grating, one determines the time delay of each temporal slice of the fundamental pulse experiences relative to the second harmonic.If the chirp rate (aperiodicity) of the QPM grating exactly matches the chirp of the input pulse, then the generated output pulse has its entire spectral component coincident in time, so the SH wave is compressed.QPM gratings generally use sign reversal of the nonlinear coefficient along the crystal length in a periodic or aperiodic fashion; i.
We defined a kind of gratings which consists of blocks of constant length of the crystal whose spontaneous polarizations, i.e. signs of the nonlinear coefficient, is arbitrary along the propagation direction.Compared with that of periodic domain inversion, this kind of structure can supply more reciprocal vector.By optimizing the domain structure, the generated second harmonic waves have the same phase for different wavelength.It is naturally expected that the second harmonic pulse can be compressed.
In our method, the length of the domain block is fixed and can be selected artificially, so the difficulty of poling can be alleviated compared with that of chirped domain inversion gratings [3][4][5][6][7][8] .Furthermore, this method can be employed when the input fundamental pulse with arbitrary amplitude and phase distribution.

Theoretical investigation
We begin the analysis with the coupled wave equations expressed in the frequency domain by considering an aperiodical QPM grating of length L with longitudinally optionally Where ˆ( , ) E z ω is the Fourier transform of the electric field , ˆ( , ) A z Ω is the frequency-envelope, and is the frequency detuning from ω 0 .These envelopes are to be distinguished from the conventional time-domain envelopes ( , ) B z t , which are defined by where ( , ) E z t is the electric field and 0 0 ( ) By the assumption of the slowly varying amplitude approximation, an undepleted pump, and a plane-wave interaction, the coupled wave equations governing the propagation of the FH envelope 1

ˆ( , )
A z Ω and SH envelope 2 ˆ( , ) A z Ω as follows (here and in the remainder of paper we use the subscript 1 to denote the FH and the subscript 2 to denote the SH); where the nonlinear polarization ˆ( , ) From the Eq.(5) -Eq.( 7), the solution of the FH envelope 1 ˆ( , ) A z Ω and SH envelope 2 ˆ( , ) A z Ω are: Where ˆ( ) ( )e x p ( ) λ is the free-space FH wavelength, and 2 n is the refractive index at the SH frequency.The It is noted that Eq. (8) and Eq. ( 9) are valid for materials with arbitrary dispersion.It is difficult to calculate 2 ˆ( , ) A L Ω for arbitrary input pulse and arbitrary material dispersion.
Assuming that GVD and higher-order dispersion terms at the FH and SH wave-pulse can be ignored, the grating can acts as a transfer function. 7,8onsidering second harmonic generation interaction 1 A Ω in Eq. ( 9), the Eq. ( 9) is simplified into

Numerical investigation and discussions
As an example, we consider the case of SHG pumped by a linearly chirped Gaussian FH pulse with optical carrier frequency 0 ω , e 1 power half-width 1 τ and real (temporal peak) amplitude 0 E .the time-domain envelope of the electric field that corresponds to this pulse is ) Using the frequency-domain envelope definition, we obtain 1 ˆ( ) A Ω for this pulse as According to Eq. (10),

ˆ( )
A Ω is determined by the modulated nonlinear coefficient ) (z d and the crystal length L , the block numbers n .In our simulations, L and n is fixed, so our target is try to find the best distribution of the signs of the nonlinear coefficient in that the generated SH pulse can be compressed.If the amplitude and phase of the continue wave component whose frequency varies among the spectra of the input Gaussian pulse are known, we will seek for the optimal distribution of the nonlinear coefficient of the grating whose phase response is the same for all frequency components of SH waves.It will lead the output SH pulse to be compressed. In our calculations, the genetic algorithms are employed to search for the best distribution of the sign of the nonlinear coefficient.In order to obtain compressed SH pulse with higher conversion efficiency and shorter pulse width.We should optimize two parameters: amplitude and phase of SH pulse.At the first simulation stage, we found that the probability to find the best distribution of nonlinear coefficients of blocks is very low when simple genetic algorithm is applied.In order to enhance the probability, an improved genetic algorithmscascading genetic algorithms is proposed.First, we use simple genetic algorithms to generate relatively good result, and save it.Redo it for several times (times need to be pre-determined).Afterwards, according to selection rules, we select proper individuals from the saved results as initial population of the simple genetic algorithms.In the final simple genetic algorithms, we adopt the elitist model, so that the best result can be handed down during the calculation.Finally, we obtain the desired result when the pre-determined calculation times reach.The flow chart of cascading genetic algorithms is shown as follows In the simulations, the parameters of the input FH pulse are taken as: 0 τ =24.0fs (FWHM is 40.0fs),C 1 =20 2 0 τ , so the e 1 power half-width 1 τ is calculated to be 481.0fs.
The center wavelength of FH is 800nm, the nonlinear optics crystal is lithium niobate and the temperature T is 25℃.The refractive index of lithium nioabte for different wavelengths are calculated from the Sellmeier data of Ref.9, the length of the grating L is 5mm, the length of each block l is chosen as 8μm, 6μm and 5μm, respectively.d(z) is optimized not only make the phase response of SH wave almost same at the output, but also make more frequencies meet QPM condition.The results are shown in Fig. 2.After optimization of domain inversion structure, the phase of generated SH becomes uniform for the wavelengths from 387nm to 413nm.The calculated results also indicate the efficiency is higher for shorter block length and more FH wavelengths can be effectively converted into SH.The synthesized SH pulses are plotted in Fig. 3.The ideal e 1 power half-width pulse (all frequencies can be converted into SH) duration for transform-limited is 17.0fs (FWHM is 28.3fs).The calculated SH pulse duration (FWHM) is 33.5fs, 32.8fs and 31.0fsrespectively, which are very close to theoretical value.Because of the limited acceptance bandwidth of the grating (lead to the spectral truncation of frequencies conversion), the SH pulse is broaden.For fixed block length l , the converting efficiency is higher when the grating length is longer.The grating length L should be properly chosen considering the converting efficiency and the group-velocity mismatch between FH and SH.N σ is normal distribution function and σ is standard deviation.By numerical simulations, we find that whenσ is set to be 0.1 m μ , the width of the generated pulse is increased by about 5% of the original one (the block length is 5 m μ ).When the value of σ is further augmented, the width of the generated pulse increases, the pulse becomes unsymmetrical and eventually the shape of the pulse becomes irregular.In an additional, the simulation results also show that the stability of the grating with shorter block length is better than those with longer block length.

Conclusions
In summary, we have theoretically demonstrated a new procedure for designing a domain inversion grating which can cause SH waves compressed during the second-harmonic generation.The improved genetic algorithm is used to optimize the distribution of the nonlinear coefficient of each block to make the phase response the same for frequencies within the spectra of the input pulse.In our numerical simulations, the generated SH wave is successfully compressed to near ideal situation.Besides pulse compression, this method can also be applied for pulse shaping with arbitrary input pulse.

2
Nonlinear optical frequency (C) 2005 OSA 5 September 2005 / Vol. 13, No. 18 / OPTICS EXPRESS 6807 describe the interacting FH and SH fields with the frequency-domain electric field envelopes, as introduced in detail in the Ref. 7. These envelopes have the frequency-dependent k-vector ) (ω k explicitly factored out and are defined by 0

Fig. 1 .
Fig. 1.The flow chart of the cascading genetic algorithms.

Fig. 3 .
Fig. 3. (a) is the input linearly chirped pulse envelope,(b), (c), (d) is the electric field of the SH, the block length of the grating is 8µ m, 6µ m and 5µm,respectively.